Explicit Expressions for Most Common Entropies

Entropies are useful measures of variation. However, explicit expressions for entropies available in the literature are limited. In this paper, we provide a comprehensive collection of explicit expressions for four of the most common entropies for over sixty continuous univariate distributions. Most of the derived expressions are new. The explicit expressions involve known special functions.

There have been several papers giving explicit expressions for entropies. Ref. [6] derived expressions for S(X) for twenty univariate distributions. Ref. [7] derived expressions for S(X) for five multivariate distributions. Ref. [8] derived expressions for S(X) and mutual information for eight multivariate distributions. Ref. [9] derived expressions for S(X) and R(X) for fifteen bivariate distributions. Ref. [10] derived expressions for S(X) and R(X) for fifteen multivariate distributions. Ref. [11] derived expressions for S(X), R(X) and the q-entropy for the Dagum distribution. Ref. [12] derived expressions for S(X) for certain binomial type distributions. Ref. [13] derived expressions for GM(X) and CE(X) for three Lindley type distributions.
All of these and other papers are restrictive in terms of the entropies considered and the number of distributions considered. In this paper, we derive expressions for (1)- (4) for more than sixty continuous univariate distributions, see Section 3. Most of the derived expressions are new. Some technicalities used in the derivations are given in Section 2. The derivations themselves are not given and can be obtained from the corresponding author. Some conclusions and future work are noted in Section 4.
the confluent hypergeometric function defined by where (a) k = a(a + 1) · · · (a + k − 1) denotes the ascending factorial; the Kummer function defined by the Gauss hypergeometric function defined by the degenerate hypergeometric series of two variables defined by the degenerate hypergeometric function of two variables defined by The properties of these special functions can be found in [14,15].

Technical Lemmas
The derivations in Section 3 use the following two lemmas. Lemma 1. The geometric mean defined by (1) can be calculated using where E(·) denotes the expectation defined by Proof. Note that Hence, the result.

Lemma 2.
The cumulative residual entropy defined by (4) can be calculated using Proof. Using the Taylor series expansion for log(1 − z), we can write Hence, the result.